Let \(\mathbf {x}\)  and \(\mathbf {y}\)  be vectors belonging to a normed linear space \(\mathcal {X}\) . We say that \(\mathbf {x}\)  is orthogonal to \(\mathbf {y}\)  in the Birkhoff-James sense [1]}, [2]} if
\(\Vert  \mathbf {x} + \beta \mathbf {y} \Vert _{\mathcal {X}} \geqslant \Vert \mathbf {x}\Vert _{\mathcal {X}}\) 
